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UNIT 5 VOCABULARY: DECIMAL NUMBERS
1.1. Decimal numbers
Decimal numbers are used in situations in which we look for more precision than whole numbers
provide.
In order to do that, we define:
• A tenth: each part that we get when we divide something in ten equal parts. It is represented
by 0.1 =
•
1
10
A hundredth: each part that we get when we divide something in a hundred equal parts. It is
represented by 0.01 =
•
1
100
A thousandth: each part that we get when we divide something in a thousand equal parts. It
is represented by 0.001 =
1
1,000
1.2. Place value
In the decimal number system ,the value of a digit depends on where it is placed:
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Adding extra zeros to the right of the last decimal digit does not change the value of
the decimal number.
For example, 3.24 = 3.240 = 3.2400 ...
1.3. How to read decimal numbers
Decimal numbers are read with each figure separate. We use a full stop (called
"point"), not a comma, before the decimal places.
You can also read the full number after the decimal point and then say the word for
the last place value.
Examples:
2.34 ➔ two point three four or 2 point thirty-four hundredths.
3.375 ➔ three point three seven five or three point three hundred and seventy-five thousandths.
0.75 ➔ (nought or zero) point seven five or seventy-five hundredths.
Exercises.
1. Read the following numbers:
120,000.321
34.76
453.897
0.54
700,560
0.054
2. Write with words the following numbers:
21.456
0.77
0.0089
5.7254
5,542.678987
8,275.4
1.4. Converting fractions into decimals
In unit 4 we studied that every fraction can be expressed as a decimal number. To convert a fraction
into a decimal, you just have to divide numerator by
denominator.
The quotient of a fraction can be:
•
•
•
Integers: no decimal part. For example,
6
=3
2
Exact (or terminating) decimals: decimal numbers that
end (or terminate).
3
5
=0.3 or
=1.25
For example,
10
4
Repeating or recurring decimal numbers: decimal numbers that have a recurring pattern of a
single or multiple digits.
1
2
31
̂
=0.333333 ...=0. 3
=0. ̂
18...
=2.58 3̂
For example,
3
11
12
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Exercise. Convert to a decimal:
1
a)
b)
4
3
20
c)
13
5
d)
17
6
e)
131
11
1.5. Converting decimals into fractions
But decimals can be expressed as fractions. To convert an exact decimal into a fraction, you have to
follow these steps:
• The numerator is formed by the digits without the decimal point.
• The denominator is the number formed by “1” and as many zeros as decimal figures the
number has.
• Reduce, if possible.
275 11
6 3
3002 1501
=
0.6= =
3.002=
=
For example, 2.75=
100 4
10 5
1,000 500
Exercise. Write these decimal numbers as fractions:
a) 0.4
b) 1.2
c) 0.045
d) 2.625
1.6. Representation of decimals on the Number Line
To represent a decimal on a number line, divide each segment of the number line into ten equal parts.
For example, to represent 8.4 on a numbe r line, divide
the segment between 8 and 9 into ten equal parts.
Likewise, to represent 8.45 on a number line, divide the
segment between 8.4 and 8.5 into ten equal parts.
Similarly, we can represent 8.456 on a number line
by dividing the segment between 8.45 and 8.46
into ten equal parts.
Exercise. Write the decimal number that the arrow points at in the following diagrams:
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2.1. Addition and subtraction of decimals
To add or subtract decimals, line up the decimal points and then follow the rules for
adding or subtracting whole numbers, placing the decimal point in the same column.
When one number has more decimal places
than another, use 0's to give them the same
number of decimal places.
For example, to add 528 and 7.49
• Line up the decimal points and add 0s on the right
of the first number.
• Then add.
2.2. Multiplication of decimal numbers
Multiplying decimals is just like multiplying whole numbers. The only extra step is to
decide how many digits to leave to the right of the decimal point. To do that, add the
numbers of digits to the right of the decimal point in both factors.
For example, to multiply 3.77 by 2.8:
Exercises. Work out:
a) 5.6 · 6.9
b) 12.37 · 76.78
c) –4.66 · 4.7
d) 0.345 · (32.4 – 4.67)
3.1. Dividing whole numbers, with decimals
To get decimals in a division, continue the whole division adding zeros to the right of the
number being divided until you get the amount of decimal digits required.
For example, divide 235:6 until the hundredth:
Exercise. Calculate with two decimal digits:
a) 56 ÷ 7
b) 7634 ÷ 34
c) – 679 : 32
d) 9783 : 127
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3.2. Dividing decimals by decimals
To divide by a decimal by another decimal, multiply the divisor by a power of 10 great
enough to obtain a whole number. Multiply the dividend by that same power of 10.
Then the problem becomes one involving division by a whole number instead
of division by a decimal.
For example:
Exercise. Calculate with three decimal digits:
a) 56.7 ÷ 2.34
b) 1432.3 ÷ 0.42
c) – 12.34 ÷ 3.5
d) 1 ÷ 1.2
3.3. Order of operations
Once again!!
When you have several operations to do, which one do you
calculate first?
We work out operations in this order:
BRACKETS
EXPONENTS (Powers, roots, etc)
DIVISION and MULTIPLICATION (working from left to right)
ADDITION and SUBTRACTION (working from left to right)
That makes
BEDMAS!
4.1. Approximations
An approximation of a number is a representation of that number that is not exact, but still close
enough to be useful.
Rounding off a decimal number to a given number of decimal places is the quickest way to
approximate a number.
For example, if you want to round off 2,6525272 to three decimal places, you would:
Step 1: Mark off the required number of decimal places. 2,652|5272
Step 2: Check the next digit to see if you must round up or round down. Remember: if the next digit is
5 or more, you must round up, and if it is 4 or less, you must round down. 2,652|5272 must be rounded
up.
Step 3: Write the final answer. 2,653 rounded to 3 decimal places.
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REMEMBER!
When the digit 5, 6, 7, 8, or 9 appears in the ones place, round up;
when the digit 0, 1, 2, 3, or 4 appears in the ones place, round down.
Exercises.
1. Round off these distances to the nearest 100 m (to one decimal place):
a) 5.768 km
c) 8.48 km
e) 17.685 km
b) 9.039 km
d) 8.41 km
f) 17.658 km
2. Round off:
a) 1.17 to the nearest tenth
d) 12.87 to the nearest unit
b) 2.375 to the nearest hundredth
e) 151.504 to the nearest hundred
c) 0.7084 to the nearest thousandth
f) 7478 to the nearest thousand
3. The height of each person has been rounded to the nearest 10 cm.
The actual heights of the six people are:
1.44 m
1.54 m
1.61 m
1.65 m
Match each person with their actual height.
1.84 m
1.85 m
4.2. Word problems
1) Ellen earns 137.40 per week. After 4 weeks, she gets an extra payment of 24.75.
She spends 354.60 in this period. How much money does she save?
2) A student studies a total time of 4 h 35 min and during this time he writes 100
min. How long, in hours, does he study without writing?
3) Susan is cooking a cake and uses 1.35kg of flour, 0.37kg of sugar, 3 eggs that
weigh 80g each and 240g of milk. What is the weight of the mixture ?
4) I buy 7 kilograms of meat and pay € 53.55 . How much does the kilogram of
meat cost?
5) Henry has € 83.40 . He buys four tickets for the cinema at € 6.50 each and 2 bags
of popcorn at € 2.25 each. How much money has he got left?
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6) A breeder gives to each pig 0.65 kg of food for every 4 kg of body weight.
There are 4 pigs of 75.8 kg, 56.4 kg, 75.4 kg and 89.3 kg. How much food must
be prepared in total?
7) In a restaurant, 7 friends are having a meal. The bill is £173.6 and each
person contributes with £25.50. What tip does the waiter receive?
8) A company produces items that are sold for £13.63 each. The daily
production is 1275 items and the cost of production is £11,324.50. What is
the daily income for the company?
9) Fernando Alonso can travel at 167.33 mph in his McLaren Formula 1 car.
How far can he go in 4 hours?
10) Ted was 88.53 cm tall when he was 3 years old. By the time he was 19, Ted had
grown a further 85.92 cm. How tall was he when he was 19?
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